Airline Fares Depend on Unobserved Route and Time Characteristics

Two Sources of Omitted Variable Bias

\[ \begin{aligned} \log(\text{fare})_{it} &= \beta_0 + \beta_1\log(\text{distance})_i + \beta_2\text{competition}_{it} + e_{it} \\ e_{it} &= \gamma_i + \delta_t + u_{it} \end{aligned} \]

Controlling for Distance and Year Dummies Does Not Remove Route Heterogeneity

The Proposed Model Falls Short

Combining FD with Year Dummies Addresses Both Sources

Two Strategies for Two-Way Heterogeneity

(1) FD + year dummies, or (2) full LSDV with dummies for both units and time periods.

Industry-Wide Patent Growth Requires Flexible Time Effects

Year Dummies Capture Any Form of Common Time Trends

Year Dummy Coefficients Measure Growth Rates

Conditional Expectations Are The Tool to Interpret

\[\begin{align*} \mathbb{E}[\log(\text{patents})_{it} \mid t=2005] &= \beta_0 + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2006] &= (\beta_0 + \beta_2) + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2007] &= (\beta_0 + \beta_3) + \beta_1\log(\text{R\&D})_{it} \end{align*}\]

• Average log change in patents across all firms
• Geometric mean growth rate of patents in the industry
  • \(\beta_2\): 2005 to 2006
  • \(\beta_3\): 2005 to 2007

Year dummies measure growth rates between periods — not “the level in 2006 vs 2005.”

Interactions Allow the R&D Elasticity to Vary Over Time

Heterogeneous Elasticities

\[ \begin{align*} \log(\text{patents})_{it} &= \beta_0 + \beta_1\log(\text{R\&D})_{it} + \beta_2\mathbb{1}[t=2006] + \beta_3\mathbb{1}[t=2007] \\ &+ \beta_4(\log(\text{R\&D})_{it} \times \mathbb{1}[t=2006]) + \beta_5(\log(\text{R\&D})_{it} \times \mathbb{1}[t=2007]) \\ &+ e_{it} \end{align*} \]

Conditional means by year

\[\begin{align*} \mathbb{E}[\log(\text{patents})_{it} \mid t=2005] &= \beta_0 + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2006] &= (\beta_0 + \beta_2) + (\beta_1 + \beta_4)\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2007] &= (\beta_0 + \beta_3) + (\beta_1 + \beta_5)\log(\text{R\&D})_{it} \end{align*}\]

Level Shifts and Slope Shifts Are Separately Identified

Decomposing Differential Effects

No need to interpret \(\beta_4\) and \(\beta_5\) individually; the conditional means do the work.

Gender Wage Gaps Changed After the Mining Boom

Three Patterns from the Data

• Gender: Men earn a constant wage premium over women
• Time trend: Wages grow over time for both groups
• Structural break (2005): After the mining company arrives, the male premium widens

The Empirical Model Interacts Gender, Time, and Post-2005

Eight Coefficients for Four Groups

\[ \begin{align*} \log(\text{wages})_{it} &= \beta_0 + \beta_1\mathbb{1}[i\text{ is male}] + \beta_2 t + \beta_3\mathbb{1}[t \geq 2005] \\ &+ \beta_4(\mathbb{1}[i\text{ is male}] \times t) + \beta_5(\mathbb{1}[i\text{ is male}] \times \mathbb{1}[t \geq 2005]) \\ &+ \beta_6(t \times \mathbb{1}[t \geq 2005]) + \beta_7(t \times \mathbb{1}[t \geq 2005] \times \mathbb{1}[i\text{ is male}]) \\ &+ e_{it} \end{align*} \]

This model captures level differences, trends, and how both changed after 2005, separately for men and women.

Conditional Means: Pre-2005

Women and Men Before the Structural Break

Women before 2005 (base category)

\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=0,t<2005] = \beta_0 + \beta_2 t \]

Men before 2005

\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=1,t<2005] = (\beta_0 + \beta_1) + (\beta_2 + \beta_4)t \]

\(\beta_1\) shifts the intercept; \(\beta_4\) shifts the slope.

Conditional Means: Post-2005

Women and Men After the Structural Break

Women after 2005

\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=0,\; t \geq 2005] = (\beta_0 + \beta_3) + (\beta_2 + \beta_6)t\]

Men after 2005

\[\begin{align*} \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=1,\; t \geq 2005] &= (\beta_0 + \beta_1 + \beta_3 + \beta_5) \\ &\quad + (\beta_2 + \beta_4 + \beta_6 + \beta_7)t \end{align*}\]

Each coefficient modifies either the intercept or slope for a specific group-period combination.

Taking Differences Isolates Each Coefficient’s Role

Condition on Group, Then Difference

Coefficient Signs Follow Directly from the Figure

Economic Interpretation

Panel Data Follows the Same Units Over Time

Definition and Structure

• \(y_{it}\), \(x_{it}\) for \(i = 1, \ldots, N\) and \(t = 1, \ldots, T\)
• Panel data: same units observed across multiple time periods
• Cross-section: one \(t\) only
• Repeated cross-section: same population, different individuals each period

Error decomposition

\[ e_{it} = \alpha_i + v_{it} \]

Panel Structure Enables Identification of Parameters

Identification vs Estimation

Example: cannot estimate returns to education via FD if education does not change over time.

Education Is Measured with Error in Both Periods

True vs Observed Variables

\[ \text{education}_{it} = \text{education}^{*}_{it} + e_{it} \]

Assumptions

• \(e_{it}\) uncorrelated with true education and other variables
• Education varies little over time for adults

Cross-Sectional Attenuation Bias Shrinks the Coefficient Towards Zero

The Baseline Problem

Population model

\[ \log(\text{wage})_i = \alpha + \beta\text{education}^{*}_i + \epsilon_i \]

Observed model substitutes \(\text{education}_i = \text{education}^{*}_i + e_i\)

Attenuation bias

\[ \text{plim}\;\hat\beta = \beta \cdot \frac{\text{Var}(\text{educ}^*)}{\text{Var}(\text{educ}^*) + \text{Var}(e)} \]

The ratio is less than 1, so the coefficient is biased towards zero (derived in Topic 6).

First-Differencing Increases the Measurement Error Variance

The Panel Data Paradox

FD of observed education

\[ \Delta\text{education}_i = \Delta\text{education}^{*}_{i} + (e_{i2} - e_{i1}) \]

If \(e_{i1}\) and \(e_{i2}\) are uncorrelated

\[ \text{Var}(e_{i2} - e_{i1}) = \text{Var}(e_{i1}) + \text{Var}(e_{i2}) \]

Roommate Nationality May Affect Student Grades

Self-Selection into Rooms Creates Endogeneity

Let \(\text{same}_{it} = \mathbb{1}[i\text{ has same-nationality roommate in } t]\)

\[ \text{grades}_{it} = \alpha + \beta\;\text{same}_{it} + e_{it} \]

• Exogeneity holds under random assignment of roommates
• If students choose roommates: an omitted equation determines room selection
• More outgoing students may prefer different nationalities and perform differently academically
• \(\text{Cov}(\text{same}_{it}, e_{it}) \neq 0\) arises from selection, not unobserved heterogeneity

Two Years of Data Require Roommate Changes

Panel Structure and Exogenous Mobility

Year 1

\(\text{grades}_{i1} = \alpha + \beta\;\text{same}_{i1} + \alpha_i + u_{i1}\)

Year 2

\(\text{grades}_{i2} = \alpha + \beta\;\text{same}_{i2} + \alpha_i + u_{i2}\)

First-differencing

\(\Delta\text{grades}_i = \beta\;\Delta\text{same}_i + \Delta u_i\)

• Critical: students must change roommates (\(\Delta\text{same}_i \neq 0\) for some)
• Exogenous mobility design — plausible if the university reassigns rooms

Random Assignment Plus Panel Data Strengthens Identification

You Get What You Pay For

• Random assignment of roommates:
  • \(\beta\) is identified even in cross-section
  • Panel adds precision: removes \(\alpha_i\) from the error \(\to\) smaller variance \(\to\) smaller standard errors
• Potential selection: panel data alone cannot solve the selection problem

First-Difference Derivation for General Panel Model

Write the model for \(t = 1\) and \(t = 2\):

\[ \begin{align*} y_{i1} &= \beta_0 + \beta_1 x_{i1,1} + \cdots + \beta_k x_{i1,k} + a_i + v_{i1} \\ y_{i2} &= (\beta_0 + \delta) + \beta_1 x_{i2,1} + \cdots + \beta_k x_{i2,k} + a_i + v_{i2} \end{align*} \]

Subtract: \(a_i - a_i = 0\).

\[ \Delta y_i = \delta + \beta_1\Delta x_{i1} + \beta_2\Delta x_{i2} + \cdots + \beta_k\Delta x_{ik} + \Delta v_i \]

The time-invariant component \(a_i\) has been eliminated. OLS on this equation yields consistent estimates under \(\mathbb{E}[\Delta v_i \mid \Delta x_{i1}, \ldots, \Delta x_{ik}] = 0\).

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Attenuation Factor: Cross-Section vs First-Differences

Cross-section

\[ \text{plim}\hat\beta_{\text{CS}} = \beta \cdot \frac{\text{Var}(\text{educ}^*)}{\text{Var}(\text{educ}^*) + \text{Var}(e)} \]

First-differences (assuming uncorrelated measurement errors)

\[ \text{plim}\;\hat\beta_{\text{FD}} = \beta \cdot \frac{\text{Var}(\Delta\text{educ}^*)}{\text{Var}(\Delta\text{educ}^*) + \text{Var}(e_{i1}) + \text{Var}(e_{i2})} \]

• Since education changes little over time:
  • \(\text{Var}(\Delta\text{educ}^*) \ll \text{Var}(\text{educ}^*)\).
  • FD \(<\) CS attenuation ratio, and the bias towards zero is worse in FD

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